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One--dimensional stellar structure and evolution programs are built using different physical prescriptions and algorithms, which means there can be variations between models' predictions even when using identical input physics. This leads to questions about whether such deviations are physical or numerical; code validation studies are important and necessary tools for studying these questions. We provide the first direct comparison between the Monash stellar evolution program and MESA for a $2M_{\odot}$ model evolved from the zero-age main sequence to the tip of the thermally pulsing asymptotic giant branch. We compare the internal structure of the two models at six critical evolutionary points and find that they are in excellent agreement in characteristics like central temperature, central density and the temperature at the base of the convective envelope during the thermally pulsing asymptotic giant branch. The hydrogen-exhausted core mass between the models differs by less than 4.2% throughout the entire evolution, the final values vary only by 1.5%. Surface quantities such as luminosity and radius vary by less than 0.2% prior to the asymptotic giant branch. During thermal pulses, the difference extends to 3.4%, largely due to uncertainties in mixing and the treatment of atmospheric boundary conditions. Given that the veteran Monash code is closed source, the present work provides the first fully open-source computational analog. This increases accessibility to precision modeling on the asymptotic giant branch and lays the groundwork for higher-mass calculations that are performed with MESA but preserve the standards of the Monash code during the AGB.